2X2 matrices can define transformations for the entire plane. In this worked example, we see how to find the image of a given vector under the transformation defined by a given matrix. Created by Sal Khan.
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- It looks like this has already been outlined in the previous section. In "Matrices as transformations of the plane".(1 vote)
- Yes, you are correct. The notation and concept of using matrices to represent transformations of vectors was already discussed in the previous section "Matrices as transformations of the plane". In that section, we saw that we can use a matrix to represent a linear transformation of a vector in a two-dimensional space.
Specifically, we can use a transformation matrix to map each point in the original space to a corresponding point in the transformed space. This transformation is performed by multiplying the transformation matrix by the column vector of the original point. The resulting column vector gives the coordinates of the transformed point in the transformed space.
In the current section, we are building upon that concept and exploring how to visualize and calculate the transformation of a specific vector using a transformation matrix. Sal uses the notation and concepts already introduced in the previous section to explain this concept.(4 votes)
- In the end, he talks about how you can verify if it ends at the point (10,9) by doing the math. Can I just do that math to find the transformation or no?(2 votes)
- When Sal talks about the transformation image, in this case [2/1 2/3] how did he get the image?(1 vote)
- In the video, the instructor is finding the image of the original vector [3 2] under the linear transformation represented by the transformation matrix:
To find the image, he first applies the transformation matrix to the unit vectors in two dimensions, i.e., the vectors [1 0] and [0 1]. The resulting vectors are the images of the unit vectors under the transformation.
The image of [1 0] is:
And the image of [0 1] is:
These two vectors form the basis for the transformed space, which is the space that contains all possible images of vectors under this transformation.
To find the image of the original vector [3 2], the instructor expresses it as a linear combination of the transformed unit vectors using the coefficients 3 and 2. That is, he writes:
[3 2] = 3[2 1] + 2[2 3]
This equation tells us that the original vector [3 2] can be expressed as a linear combination of the transformed unit vectors.
The image of the original vector is then obtained by adding up the scaled images of the unit vectors, which gives:
[3 2] (image) = 3[2 1] + 2[2 3] = [10 9]
So the image of the original vector [3 2] under this transformation is the vector [10 9].(2 votes)
- I don't understand what is meant by the word "image"(1 vote)
- In linear algebra, the term "image" usually refers to the output of a linear transformation. When we apply a linear transformation to a vector or a set of vectors, we get a new vector or set of vectors as the output, which is called the "image" of the original vector or set of vectors.
In the context of this video, the instructor is using the term "image" to refer to the output obtained by applying the transformation matrix to the unit vectors in two dimensions. These image vectors are then used to find the image of the original vector after the transformation.
So, in short, "image" refers to the output obtained by applying a linear transformation to a vector or a set of vectors.(1 vote)
- I'm curious about what notation you use when you are doing a transform for matrices(1 vote)
- When performing a transformation with matrices, I would typically use the standard mathematical notation. The notation for matrix multiplication is the dot product, where the dot represents the multiplication operation. For example, if we have a matrix A and a vector x, we would write the matrix-vector multiplication as Ax.
When we perform a transformation using a matrix, we apply the matrix to the vector that we want to transform. This can be done by multiplying the matrix and the vector together, which gives us a new vector that represents the transformed version of the original vector.
In the video, Sal uses a similar notation for the transformation by writing the image of the original vector as a linear combination of the images of the unit vectors under the transformation matrix. He denotes this with a prime symbol to indicate that it is the transformed vector, and he uses the same dot product notation to compute the transformation.(1 vote)
- [Instructor] Let's say that we have the vector three, two. We know that we can express this as a weighted sum of the unit vectors in two dimensions or we can view it as a linear combination. And you could view this as three times the unit vector in the X direction which is one zero plus two times the unit vector in the Y direction, which is zero, one. And we can graph three, two by saying okay, we have three unit vectors in the X direction. This would be one right over there. That would be two and then that would be three. And then we have plus two unit vectors in the Y direction. So one and then two, and then we know where our vector is or what it would look like. The vector three, two would look like this. Now let's apply a transformation to this vector. And so let's say we have the transformation matrix. I'll write it this way. Two, one, two, three. Now we've thought about this before. One way of thinking about a transformation matrix is it gives you the image of the unit vectors. And so instead of being this linear combination of the unit vectors, it's going to be this linear combination of the images of the unit vectors when we take the transformation. What do I mean? Well, instead of having three one, zeros, we are now going to have three two, ones. Instead of having two zero, ones, we're now going to have two two, threes. So I could write it this way. Let me write it this way, the image of our original vector. I'll put a prime here to say we're talking about its image is going to be three times instead of one, zero, it's going to be times two, one vectors. That's the image of the one, zero unit vector under this transformation. And then we're gonna say plus two instead of zero, one, we're gonna look at the image under the transformation of the zero, one vector, which the transformation matrix gives us and that is the two, three, vector. Two, three, and we can graph this. If we have three two, ones and two two, threes, what I could do is overlay this extra grid to help us. So this is two, one, that's one to one, that is two two, ones going here. And then we have three two, ones right over here. So there's three two, ones. Let me do this in this color. This part right over here is going to be this vector. The three two, ones is going to look like that. And then to that, we add two two, threes. So this is going to... Let's see, two and then three so this is going to be one two, three, and then we have two two, threes. So we end up right over there. And so let me actually get rid of this grid so we can see things a little bit more clearly. And so we have here in purple, we have our original three, two vector and now the image is going to be three two, ones plus two two, threes so the image of our three, two vector under this transformation is going to be this vector that I'm drawing right here. And it looks when I eyeball it, it looks like it is the 10, nine vector and we can verify that by doing the math right over here. So let's do that. This is going to be equal to three times two is six, three times one is three. And we're going to add that to two times two is four, two times three is six. And indeed you add the corresponding entry, six plus four is 10 and three plus six is nine. And we're done. The important takeaway here is that any vector can be represented as a linear combination of the unit vectors. Now when we take the transformation, it's now going to be a linear combination not of the unit vectors, but of the images of the unit factors. And we saw that visually and we verified that mathematically.